e ective potential for relativistic...

Effective potential for relativistic scattering

Janos Balog

Wigner Research Centre, Budapest and CAS IMP, Lanzhou

in collaboration with

Pengming Zhang Mahmut Elbistan

CAS IMP, Lanzhou CAS IMP, Lanzhou

• Quantum Inverse Scattering

• Nucleon potential from lattice QCD

• Sine-Gordon model

• Sine-Gordon effective potential from QIS

Szegedi Tudomanyegyetem, 2017 marcius 16

Quantum Inverse Scattering

Szegedi Tudomanyegyetem, 2017 marcius 16 1

Inverse scattering problems

Inverse scattering: find potential from scattering data

All physics is inverse scattering:

Newton’s law

Rutherford’s experiment

Watson & Crick double helix

Direct scattering: given potential (forces) find scattering data

Inverse scattering: given scattering data find forces

Szegedi Tudomanyegyetem, 2017 marcius 16 2

1-dimensional quantum mechanics on the half-line

Schrodinger operator: `u = −u′′ + qu

self-adjoint Hamiltonian: (~, m etc. scaled out)

`u = −u′′ + qu and boundary condition u(0) = 0

q(x) potential x ≥ 0

q(x) ∼ p(p−1)x2 x→ 0 p > 1

u(x) ∼ xp regular solution

u(x) ∼ x1−p singular solution

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1 2 3 4 5

5

10

15

20

Figure 1: The singular potential q(x)

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Three solutions of the `u = k2u Schrodinger equation:

φ(x, k) ∼ xp x→ 0 physical solution

φ(x, k) ∼ x1−p x→ 0 singular solution

f(x, k) ∼ eikx x→∞ Jost solution

f∗(x, k) = f(x,−k)

Jost function f(k):

f(x, k) = f(k)φ(x, k) + f(k)φ(x, k)

φ(x, k) = 2p−12ik {f(−k)f(x, k)− f(k)f(x,−k)}

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2 4 6 8 10

2

4

6

Figure 2: The singular potential q(x) and the regular wave function φ(x, k) at k = 1.5

The constant total energy is also shown.

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phase shift δ(k):

f(k) = |f(k)|e−iδ(k)

S-“matrix”:

S(k) = f(−k)f(k) = e2iδ(k)

asymptotics of the physical solution:

φ(x, k) ∼ −2p−12ik f(k)

{e−ikx − S(k)eikx

}

∼ sin [kx+ δ(k)]

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(Solvable) example

q(x) = p(p−1)

sinh2(x)

change of variables:

u(x) = eikxF (z) z = 11−e−2x

hypergeometric equation:

z(1− z)F ′′(z) + [c− (a+ b+ 1)z]F′(z)− abF (z) = 0

a = p b = 1− p c = 1 + ik

hypergeometric function: 2F1(a, b, c; z)

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physical solution:

φ(x, k) = 12p

(1− e−2x

)peikx2F1

(p, p− ik, 2p; 1− e−2x

)

Jost solution:

f(x, k) =(1− e−2x

)peikx2F1

(p, p− ik, 1− ik; e−2x

)

Jost function:

f(k) = 12p−1

Γ(1−ik)Γ(2p−1)Γ(p)Γ(p−ik)

S-matrix:

S(k) = Γ(1+ik)Γ(p−ik)Γ(1−ik)Γ(p+ik)

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high energy asymptotics:

δ(k) = π2(1− p)− d1

k + . . . δ(∞) = π2(1− p)

p = 2:

f(k) = 11−ik S(k) = 1−ik

1+ik

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Inverse scattering in three steps

• step 1: scattering data

F (x) = 12πix

∫ ∞

−∞dk eikx S′(k)

• step 2: Marchenko equation

F (x+ y) +A(x, y) +

∫ ∞

x

dsA(x, s)F (s+ y) = 0

• step 3: potential

q(x) = −2 ddxA(x, x)

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p = 2 example

• step 1: scattering data

F (x) = −2 e−x

• step 2: Marchenko equation

A(x, y) = e−y

sinh(x)

• step 3: potential

A(x, x) = coth(x)− 1 q(x) = −2 ddxA(x, x) = 2

sinh2(x)

Szegedi Tudomanyegyetem, 2017 marcius 16 12

Nucleon potential from first principles

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Modern nucleon-nucleon potential

Origin of the N-N Repulsive Core

The Most Fundamental Problem in Nuclear Physics

!!!!!!

r

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!!""#""'()"*)"+)",

!!!"#""-./01"2

(Taketani)

IIIIII

Long range part

one pion exchange potential(OPEP)

I

II Medium range part

σ, ρ, ω exchange2π exchange

III Short range part

repulsive core (RC)

quark ?Bonn: Machleidt, Phys.Rev. C63(Ô01)024001Reid93: Stoks et al., Phys. Rev. C49(Ô94)2950.AV18: Wiringa et al., Phys.Rev. C51(Ô95) 38.

R. Jastrow(1951)

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Szegedi Tudomanyegyetem, 2017 marcius 16 15

Our strategy in lattice QCD

Consider “elastic scattering”

NN → NN NN → NN + others (NN → NN + π,NN + NN, · · ·)

Elastic threshold

• S-matrix below inelastic threshold. Unitarity gives

Quantum Field Theoretical consideration

S = e2iδ

Full details: Aoki, Hatsuda & Ishii, PTP123(2010)89.

energy Wk = 2√k2 +m2

N < Wth = 2mN +mπ

Szegedi Tudomanyegyetem, 2017 marcius 16 16

Step 1

define (Equal-time) Nambu-Bethe-Salpeter (NBS) Wave function

ϕk(r) = 〈0|N(x+ r, 0)N(x, 0)|NN,Wk〉QCD eigen-state

N(x) = εabcqa(x)qb(x)qc(x): local operator

Spin model: Balog et al., 1999/2001

“scheme”

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partial wave

r = |r| → ∞

scattering phase shift (phase of the S-matrix by unitarity) in QCD !

ϕlk → Al

sin(kr − lπ/2 + δl(k))

kr

Asymptotic behavior of NBS wave function Lin et al., 2001; CP-PACS, 2004/2005

cf. Luescher’s finite volume method

no interaction

interaction range

L

allowed k at L δl(kn)

NBS wave function scattering wave function in quantum mechanics

Szegedi Tudomanyegyetem, 2017 marcius 16 18

Step 2

• Define a potential through application of a Schrodinger operator:

Vk(r) =

[Ek+

1µ∇

2]ϕk(r)

ϕk(r)

where Ek = k2

µ is the kinetic energy. Note the energy dependence of potential!

Step 3

• Solve the Schrodinger equation with this (zero energy) potential in infinite volume to find phase

shifts and possible bound states below the inelastic threshold.

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Qualitative features of NN potential reproduced!

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T. Hatsuda N. Ishii

1st paper: Ishii-Aoki-Hatsuda, PRL 90(2007)0022001

has been selected as one of 21 papers in Nature Research Highlights 2007. (One from Physics, Two from Japan, the other is on “iPS” by Sinya Yamanaka et al. )

“The achievement is both a computational tour de force and a triumph for theory.”

RVFODIFE

(courtesy of Sinya Aoki)

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1S0

aexp0 (1S0) = 23.7 fm

a0(1S0) = 1.6(1.1) fm

(courtesy of Sinya Aoki)

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Problems with the NBS approach

• Dependence of potential on choice of nucleon operator

• Energy dependence of the NBS potential

Possible solutions:

define nonlocal, but energy-independent potential

define the zero-momentum potential

Uo(r) = limk→0

V NBSk (r)

correctly reproduces scattering lengths, but effective range different

Szegedi Tudomanyegyetem, 2017 marcius 16 22

Sine-Gordon model

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Sine-Gordon model

Very well known: RFT (quantum/classical)

RM (quantum/classical) [alias RS model]

Integrable (solvable) =⇒ (almost) everything calculable

spectrum (solitons, anti-solitons, breathers)

S-matrix

Form-factors, correlators, free energy, ...

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Lagrangian (~ = c = 1 RQFT conventions)

L = 12φ

2 − 12φ′2 + µ2

β2 cos(βφ)

SG coupling β: (equivalence to Thirring model)

0 < β <√

8π β = 2√π FF point

equations of motion: (Kl)Sine-Gordon equation

2ϕ+ µ2 sinϕ = 0 (ϕ = βφ)

parameters:

p = 4πβ2 ν = 1

2p−1

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soliton mass:

m = 2p−1π µ

bound states (breathers):

mk = 2m sin(π2νk

)k = 1, 2, · · · < 2p− 1

Ruijsenaars-Schneider RQM description: zero-momentum potential

qo(x) = 4sinh2(πνx)

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Sine-Gordon S-matrix

Rapidity parametrization: θ = θ1 − θ2

pi = mc sinh(θi) Ei = mc2 cosh(θi) Ei =√

(mc2)2 + (pic)2

soliton-soliton S-matrix (no bound states):

Σ(θ) = exp

{i

∫ ∞

0

dωω sin

(2πθω

) sinh((ν−1)ω)cosh(ω) sinh(νω)

}

phase shift:

Σ(θ) = e2iδ(θ) δ(∞) = π2(1− p)

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for integer p explicit:

Σ(θ) =

p−1∏

m=1

sm−i sinh(θ)sm+i sinh(θ) sm = sin(mνπ)

2-particle scattering

• −→ p1 p2 ←− •x1 x2

initially: p1 > p2 x2 > x1 all times

asymptotic wave function (x2 − x1)→∞:

Φ(x1, x2) ≈ ei(k1x1+k2x2) + S(p1, p2)ei(k2x1+k1x2)

ki = pi~ i = 1, 2 wave numbers

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Relativistic:

SR(p1, p2) = −Σ(θ1 − θ2) θi = arcsinh(pimc

)

NR Schrodinger equation:

HΦ = EΦ H = − ~2

2m∂2

∂x21− ~2

2m∂2

∂x22

+ U(x2 − x1)

separating COM and relative motion:

Φ(x1, x2) = eiK(x1+x2) Ψ(x2 − x1)

effective 1-particle Schrodinger equation:

− ~2

m Ψ′′(x) + U(x)Ψ(x) = ~2

mκ2Ψ(x)

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total energy:

E = ~2

2m(k21 + k2

2) = ~2

m (K2 + κ2)

k1 = K + κ k2 = K − κasymptotic wave function:

Ψ(x) ≈ −A(κ)eiκx + e−iκx x→∞

NR S-matrix:

SNR(p1, p2) = −A(p1−p2

2~)

= −S(p1−p2mc

)

rescaling between physics ←→ maths conventions:

A(κ) = S(2κL) L = ~mc Compton length

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Identification?

SNR(p1, p2) ∼ SR(p1, p2)

S(p1−p2mc

)∼ Σ

(arcsinh

(p1mc

)− arcsinh

(p2mc

))

2 special cases of interest:

I (fixed target) p2 = 0 p1 = kmc

SI(k) = Σ (arcsinh(k))

II (centre of mass) p1 + p2 = 0 p1 − p2 = kmc

SII(k) = Σ (2 arcsinh(k/2))

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Effective Sine-Gordon potential

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effective Sine-Gordon potentials

Sine-Gordon NR S-matrix, I detemination:

SI(k) =

p−1∏

m=1

sm−iksm+ik sm = sin(mνπ)

Sine-Gordon NR S-matrix, II detemination:

SII(k) =

p−1∏

m=1

sm−ik√

1+k2/4

sm+ik√

1+k2/4sm = sin(mνπ)

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simplest case: I; p = 2

SI(k) = s1−iks1+ik = 1−ik/s1

1+ik/s1s1 = sin

(π3

)=√

32

rescaling of basic p = 2 case:

qI(x) = 32

1

sinh2(√

32 x

)

zero-momentum potential:

qo(x) = 4

sinh2(π3x)

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Inverse scattering case I, general p

step1: Fourier transformation

F (x) = −p−1∑

m=1

Rm e−smx

residues:

Rm = 2sm∏

n 6=m

sn+smsn−sm

step2: Marchenko equation

Ansatz:

A(x, y) =

p−1∑

m=1

Rm bm(x) e−sm(x+y)

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Marchenko equation reduced to system of algebraic equations:

bm = 1 +

p−1∑

n=1

znbnsm+sn

zm(x) = Rm e−2smx

step3: calculation of potential

A(x, x) =

p−1∑

m=1

bm(x)zm(x)

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p = 3 solution

A(x, x) = −s1 − s2 +(s2

1−s22) sinh(s1x) sinh(s2x)

D(x)

determinant:

D(x) = s2 cosh(s2x) sinh(s1x)− s1 cosh(s1x) sinh(s2x)

short distance:

A(x, x) ≈ 3x

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Results

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1 2 3 4 5 6

5

10

15

20

Figure 3: Comparison of the integrated effective potential A(x, x) (solid) and the corresponding

zero-momentum Ao(x, x) (dashed) for p = 3.

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1 2 3 4 5 6

10

20

30

40

Figure 4: Comparison of the integrated effective potential A(x, x) (solid) and the corresponding

zero-momentum Ao(x, x) (dashed) for p = 4.

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0.5 1 1.5 2

2

4

6

8

10

12

Figure 5: The integrated effective potential in the COM frame for p = 2 (dots). For comparison

the analytically obtained LAB frame integrated effective potential A(x, x) (solid) is also shown.

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0.5 1 1.5 2

10

20

30

40

Figure 6: The integrated effective potential in the COM frame for p = 3 (dots). For comparison

the analytically obtained LAB frame integrated effective potential A(x, x) (solid) is also shown.

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Figure 7: Comparison of integrated SG effective potentials for p = 3. The solid (red) line, the

(blue) dots and the dashed (black) line are the LAB frame, the COM frame and the zero-momentum

potential, respectively.

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Work in progress

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SO FAR

• SI(k) =⇒ qI(x) p = 2, 3, 4, 5

• SII(k) =⇒ qII(x) [numerically] p = 2, 3, 4

• compare qI(x), qII(x), qo(x)

TO DO

• qI(x) general p?

• QIS with incomplete data?

• method can be used in nuclear physics?

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Thank you!

Szegedi Tudomanyegyetem, 2017 marcius 16 46